sequence (x+1)^x + x^x

[Jim Nastos]

On Sat, 23 Aug 2003, cino hilliard wrote:

> a(n) =  (x+1)^x + x^x x=1, x=2,4,6,n for  n even
 (a() is prime)

> I submitted this sequence having computed only 3 terms(contrary to the 
> rules)  thinking for sure it has
> more terms than 3,13,881.

  As another reader has pointed out, primes can be very sparse when you 
get into large numbers, so finding another is possible but unlikely. Using 
Maple, no more primes have been found up to x= 704  (note that (x+1)^x+x^x 
has over 2000 digits for this value.)

>      print1(3" ");
>      forstep(x=2,n,2,
>        y=(x+1)^x+x^x;
>         if(isprime(y),print1(y" "))
> 
> Maybe someone can further limit the search or find more terms?

  Since primality testing is costly for large numbers, you might want to 
verify that y is +/- 1 modulo 6 before testing primality as any prime > 3 
must satisfy that. But since your number y is always going to be odd, the 
only way it won't be +/- 1 mod 6 is if it is a multiple of 3 so I suppose 
this is equivalent to just checking divisibility by 3 before testing 
primality. I'm not sure if that's useful.

Jim